dx/sin(x)=-d(cosx)/sin^2(x)=d(cos(x))/(cos^2(x)-1)
=-1/2 d(cos(x))/(1+cos(x)) -1/2 d(cos(x))/(1-cos(x))
=-1/2 log(1+cos(x))+1/2 log(1-cos(x))
=1/2 log((1-cos(x))/(1+cos(x))
=-1/2 d(cos(x))/(1+cos(x)) -1/2 d(cos(x))/(1-cos(x))
=-1/2 log(1+cos(x))+1/2 log(1-cos(x))
=1/2 log((1-cos(x))/(1+cos(x))